How Computers Store Numbers
Computer systems are constructed of digital electronics. That means that their electronic circuits
can exist in only one of two states: on or off. Most computer electronics use voltage levels to
indicate their present state. For example, a transistor with five volts would be considered "on",
while a transistor with no voltage would be considered "off." Not all computer hardware uses
voltage, however. CD-ROM's, for example, use microscopic dark spots on the surface of the disk
to indicate "off," while the ordinary shiny surface is considered "on." Hard disks use
magnetism, while computer memory uses electric charges stored in tiny
capacitors to indicate "on" or "off."
These patterns of "on" and "off" stored inside the computer are used to
encode numbers using the binary number system. The binary number system is
a method of storing ordinary numbers such as 42 or 365 as patterns of 1's and
0's. Because of their digital nature, a computer's electronics can easily manipulate numbers
stored in binary by treating 1 as "on" and 0 as "off." Computers have circuits that can add,
subtract, multiply, divide, and do many other things to numbers stored in binary.
How Binary Works
The decimal number system that people use every day contains ten digits, 0 through 9. Start
counting in decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Oops! There are no more digits left. How do we
continue counting with only ten digits? We add a second column of digits, worth ten times the
value of the first column. Start counting again: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (Note
that the right column goes back to zero here.), 21, 22, 23, ... , 94, 95, 96, 97, 98, 99, Oops! Once
again, there are no more digits left. The only way to continue counting is to add yet another
column worth ten times as much as the one before. Continue counting: 100, 101, 102, ... 997,
998, 999, 1000, 1001, 1002, .... You should get the picture at this point.
Another way to make this clear is to write decimal numbers in expanded notation. 365, for
example, is equal to 3×100 + 6×10 + 5×1. 1032 is equal to 1×1000 + 0×100 + 3×10 + 2×1. By
writing numbers in this form, the value of each column becomes clear.
The binary number system works in the exact same way as the decimal system, except that it
contains only two digits, 0 and 1. Start counting in binary: 0, 1, Oops! There are no more binary
digits. In order to keep counting, we need to add a second column worth twice the value of the
column before. We continue counting again: 10, 11, Oops! It is time to add another column
again. Counting further: 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111....
Watch the pattern of 1's and 0's. You will see that binary works the same way decimal does, but
with fewer digits.
Binary uses two digits, so each column is worth twice the one before. This fact,
coupled with expanded notation, can be used convert between from binary to
decimal. In the binary system, the columns are worth 1, 2, 4, 8, 16, 32, 64, 128,
256, etc. To convert a number from binary to decimal, simply write it in
expanded notation. For example, the binary number 101101 can be rewritten
in expanded notation as 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1. By simplifying
this expression, you can see that the binary number 101101 is equal to the
decimal number 45.
An easy way to convert back and forth from binary to decimal is to use Microsoft Windows
Calculator. You can find this program in the Accessories menu of your Start Menu. To perform
the conversion, you must first place the calculator in scientific mode by clicking on the View
menu and selecting Scientific mode. Then, enter the decimal number you want to convert and
click on the "Bin" check box to convert it into binary. To convert numbers from binary to
decimal, click on the "Bin" check box to put the calculator in binary mode, enter the number, and
click the "Dec" check box to put the calculator back in decimal mode.
Representation
Binary numbers and arithmetic let you represent any amount you want using
just two digits: 0 and 1. Here are some examples:
Decimal 1 is binary 0001
Decimal 3 is binary 0011
Decimal 6 is binary 0110
Each digit "1" in a binary number represents a power of two, and each "0"
represents zero:
0001 is 2 to the zero power, or 1
0010 is 2 to the 1st power, or 2
0100 is 2 to the 2nd power, or 4
1000 is 2 to the 3rd power, or 8.
When you see a number like "0101" you can figure out what it means by
adding the powers of 2:
0101 = 0 + 4 + 0 + 1 = 5
1010 = 8 + 0 + 2 + 0 = 10
0111 = 0 + 4 + 2 + 1 = 7
Larger Numbers
Here are the numbers from 0 to 15, in binary:
0000 = 0
0001 = 1
0010 = 2
0011 = 3
0100 = 4
0101 = 5
0110 = 6
0111 = 7
1000 = 8
1001 = 9
1010 = 10
1011 = 11
1100 = 12
1101 = 13
1110 = 14
1111 = 15
To represent bigger whole numbers (integers), you need more bits -- more
places in the binary number:
10000101 = 128 + 0 + 0 + 0 + 0 + 4 + 0 + 1 = 133.
That was 8 bits. 8 bits is usually called a "byte", and it's the size usually used
to represent an alphabetic character -- "A" is 65, or 01000001
Some other terms you'll hear are:

kilobyte, which is 1024 bytes (1024 is 2 to the 10th power)

megabyte, which is roughly a million bytes.
Typical sizes for personal computer RAM (random access memory) are 512 to 1024 megabytes,
while hard disks now start around 40 gigabytes. Since each byte can represent one character of
the alphabet, that means a hard disk might hold something like 80000 million characters, or
15000 million words of "raw" text. Documents formatted in a word processor take up a lot more
space, though, and the operating system and software usually fill at least 100 megabytes.
To represent real numbers, fractions, or very large numbers, binary systems use "floating point
arithmetic." That's another topic.
Why Use Them?
For computers, binary numbers are great stuff because:

They are simple to work with -- no big addition tables and
multiplication tables to learn, just do the same things over and over,
very fast.

They just use two values of voltage, magnetism, or other signal, which
makes the hardware easier to design and more noise resistant.
ASCII Table
Decimal
------048
049
050
051
052
053
054
055
056
057
065
066
067
068
069
070
071
072
Octal
----060
061
062
063
064
065
066
067
070
071
101
102
103
104
105
106
107
110
Hex
--030
031
032
033
034
035
036
037
038
039
041
042
043
044
045
046
047
048
Binary
-----00110000
00110001
00110010
00110011
00110100
00110101
00110110
00110111
00111000
00111001
01000001
01000010
01000011
01000100
01000101
01000110
01000111
01001000
Value
----0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
G
H
073
074
075
076
077
078
079
080
081
082
083
084
085
086
087
088
089
090
097
098
099
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
111
112
113
114
115
116
117
120
121
122
123
124
125
126
127
130
131
132
141
142
143
144
145
146
147
150
151
152
153
154
155
156
157
160
161
162
163
164
165
166
167
170
171
172
049
04A
04B
04C
04D
04E
04F
050
051
052
053
054
055
056
057
058
059
05A
061
062
063
064
065
066
067
068
069
06A
06B
06C
06D
06E
06F
070
071
072
073
074
075
076
077
078
079
07A
01001001
01001010
01001011
01001100
01001101
01001110
01001111
01010000
01010001
01010010
01010011
01010100
01010101
01010110
01010111
01011000
01011001
01011010
01100001
01100010
01100011
01100100
01100101
01100110
01100111
01101000
01101001
01101010
01101011
01101100
01101101
01101110
01101111
01110000
01110001
01110010
01110011
01110100
01110101
01110110
01110111
01111000
01111001
01111010
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